We work in an office which does not have rooms but instead has compartments which are split by dividing walls and have open tops. The walls are at shoulder height. What are these compartments called in business environment?

This is the correct answer. The open-topped compartments used to create a small amount of semi-personal, semi-private space for office workers are called "cubicles". There are many varieties of cubicles, mainly varying based on the height of the walls (or dividers). Many western companies use walls of 6ft or higher, many eastern companies tend to use lower walls. And there are variations depending on the purpose. A help desk where you need lots of collaboration may use lower walls even in a western company.
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music2myearDec 30 '11 at 15:37

We just call them cubes for short. I'm in a newer LEED-certified building so the walls are only about 5 feet tall and the top foot is opaque glass to allow sunlight in.
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DaveJan 2 '12 at 18:25

One space is a cubicle; the entire area is a cube farm.
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Monica CellioDec 30 '11 at 15:57

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Note though that "cube farm" is pretty disparaging. It'd be perfectly fine to use the word cubicle when talking to HR or upper management, but you'd be taking your job into your hands if you were to use the phrase cube farm with them.
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MarthaªDec 30 '11 at 17:37

The whole area is indeed called an open-plan office, but OP is asking what the individual sections are called.
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FumbleFingersDec 30 '11 at 15:19

Open-plan office is more general; it can include expanses of desks without the dividers (or with dividers only on one or two sides, while a cubicle usually has 3 walls).
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Monica CellioDec 30 '11 at 17:26

When I worked at the Bodleian Law Library in Oxford many years ago, there was a series of small cubicles reserved for senior members of the Faculty of Law.These cubicles were called "carrels". It is the only time I have every heard this word used. I don't know whether it could be applied to the case in question.

In business use, a carrel is a desk with attached dividers, whereas a cubicle is an enclosed space that can contain a desk (or contain carrels, I suppose). I think the terms are used more interchangeably in academia, as a "carrel" at my alma mater's library was any semi-private study area in the library, some of which were desks, others recessed cells, and others akin to small offices, albeit with screens instead of walls.
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chosterJan 3 '12 at 20:22

workstation: "an area with equipment for the performance of a specialized task usually by a single individual", cubicle: "a small partitioned space; especially : one with a desk used for work in a business office" -- Merriam-Webster
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NikitaDec 30 '11 at 15:41